2. SOLUTION AND DISCUSSION

Stacking model (1) over the observations in the first subsample, we have

where

Similarly, stacking model (1) over the observations in the second subsample, we have

where the notation is defined similarly to before. Finally, stacking (2a) over (2b), we have

where

Under assumptions (A1)–(A3), we can estimate β in (3) (or (1)) by GLS using the first subsample alone or using the entire sample. Thus, we define the following two GLS estimators of β:

where V₁₁ = var(ε₁).

Gourieroux and Monfort (1980, p. 1093) derived the necessary and sufficient condition for

to have the same finite-sample efficiency (i.e., the second subsample is fully superfluous for the efficient estimation of β):

In this note, we wish to extend their result to a subvector of β. More specifically, we seek to find necessary and sufficient conditions for the second subsample to be partially superfluous for the BLUE of β₁ alone.

Under assumptions (A1)–(A3), it is well known that

is BLUE with

is also unbiased with

. Notice that β₁ = Sβ, where the selection matrix S is defined as S = (I_K1, 0_K1×K2). We can then easily obtain

Because

is the BLUE for β in (1),

is always positive semidefinite and equals zero, if, and only if,

, or equivalently,

There are several ways to find the necessary and sufficient conditions for equation (7) to be true; for example, one obvious but “brute-force” approach is to use the partitioned matrix inverse rule for (X′V⁻¹X)⁻¹ and then simplify the resulting expression. In this note, we will use a much simpler and relatively novel approach to finding the necessary and sufficient conditions for (7) to hold; more specifically, we will calculate the rank of the matrix on the left-hand side of (7). For this purpose, we first recall a simple matrix fact: a matrix is a zero matrix if, and only if, its rank is zero. Thus, (7) holds if, and only if,

To find the rank of the left-hand side of (8), we need to use two well-known rank formulas for partitioned matrices due to Marsaglia and Styan (1974).

LEMMA 1. (i) Let A be a nonsingular matrix of order m, let B, C, and D be m × k, l × m, and l × k matrices, respectively. Then

(ii) If M is symmetric positive semidefinite, then

It can now be easily seen from (9) that CA⁻¹B = D if and only if rk(D − CA⁻¹B) = 0, or equivalently

. In fact, any matrix expression involving inverse of matrix can always be rewritten in the form of the Schur complement D − CA⁻¹B. Then, using (9), we can construct the corresponding partitioned matrix consisting of submatrices A, B, C, and D. When the resulting partitioned matrix can be simplified by elementary block matrix operations, we can obtain some nontrivial rank formula for D − CA⁻¹B, and, as a consequence, we can further derive necessary and sufficient conditions for CA⁻¹B = D to be true.

Given Lemma 1, we are now ready to derive the necessary and sufficient conditions for the second subsample to be partially superfluous for the BLUE of β₁. We first establish Lemma 2, as follows.

LEMMA 2. Partition the covariance matrix σ²V into a 2 × 2 block matrix, with the (i,j)-block V_ij = E(ε_iε_j′) for i,j = 1,2. Then,

Proof. See the Appendix.

Now, using Lemma 2 and the fact that a matrix is a zero matrix if and only if its rank is also equal to zero, we can readily obtain the main result of this note.

THEOREM 3. Under assumptions (A1)–(A3), given the first subsample, the second subsample is partially superfluous for the BLUE of β₁ in model (1) if, and only if,

One way to provide an intuition for the necessary and sufficient condition (12) is as follows. Premultiplying (3) by

, we have

or simply written as

where e₂ = ε₂ − V₂₁V₁₁⁻¹ε₁ is the linear projection error of ε₂ onto ε₁ and thus is uncorrelated with ε₁. Now, premultiplying (13b) again by

with M₂ = I_n − X₁₂*(X₁₂*′X₁₂*)⁻¹X₁₂*′, we have

where y₂** = y₂* − X₂₂*(X₁₂*′X₁₂*)⁻¹X₁₂*′y₁*, X₂₁** = X₂₁* − X₂₂*(X₁₂*′X₁₂*)⁻¹X₁₂*′X₁₁*, and e₂** = e₂ − X₂₂*(X₁₂*′X₁₂*)⁻¹X₁₂*′ε₁*. Here we particularly notice that the two regression errors in (14), M₂ε₁* and e₂**, are uncorrelated as a result of cov(e₂, ε₁*) = 0, var(ε₁*) = σ²I_n, and X₁₂*′M₂ = 0. Then, if the error vector of the original regression model (3), (ε₁′, ε₂′)′, is distributed as multivariate normal, the conditional distribution of y₂** given y₁ and X is also multivariate normal, its covariance matrix does not depend on β₁, and its conditional mean is X₂₁** β₁. Thus, by the definition of y₂** = (y₂ − V₂₁V₁₁⁻¹y₁) − X₂₂*(X₁₂*′X₁₂*)⁻¹X₁₂*′V₁₁^−1/2y₁, we can deduce that the conditional distribution of y₂ given y₁ and X is also normal, its covariance matrix does not depend on β₁, and its conditional mean is V₂₁V₁₁⁻¹y₁ + X₂₂*(X₁₂*′X₁₂*)⁻¹X₁₂*′V₁₁^−1/2y₁ + X₂₁** β₁, which does not depend on the unknown parameter vector β₁, if, and only if, X₂₁** = 0. Now, let A = A₂A₁ and Q = I. Then, the linear structure (A,Q) is sufficient for the BLUE of β₁ in the sense of Gourieroux and Monfort (1980) if, and only if, X₂₁** = 0, which, by the definition of X₂₁** in (14), is just the necessary and sufficient condition given in Theorem 3. This conditional distribution interpretation of the partially superfluous condition (12) of Theorem 3 is in principle the same as the approach used by Gourieroux and Monfort (1980, p. 1093) for deriving their necessary and sufficient condition for the second subsample to be fully superfluous for the BLUE of the whole parameter vector β.

Remarks.